Question

example 1
a county superintendent of highways is interested in the numbers of different types of vehicles that regularly travel within his county. in the month of august, a total of 192 registrations were purchased for passenger cars and pickup trucks at the local department of motor vehicles (dmv). the dmv reported that in the month of august, for every 5 passenger cars registered, there were 7 pickup trucks registered. how many of each type of vehicle were registered in the county in the month of august?
a. using the information in the problem, write four different ratios and describe the meaning of each.
b. make a tape diagram that represents the quantities in the part-to-part ratios that you wrote.
c. how many equal-sized parts does the tape diagram consist of?
d. what total quantity does the tape diagram represent?

Explanation:

Part a

Step 1: Identify the given ratio

We know that for every 5 passenger cars, there are 7 pickup trucks. So the ratio of passenger cars to pickup trucks is \(5:7\). This means that for every 5 units of passenger cars, there are 7 units of pickup trucks.

Step 2: Find the reciprocal ratio

The ratio of pickup trucks to passenger cars is \(7:5\). This represents the number of pickup trucks relative to the number of passenger cars.

Step 3: Find the ratio of passenger cars to total vehicles

First, find the total number of parts in the ratio \(5 + 7=12\). So the ratio of passenger cars to total vehicles is \(5:12\). This means that passenger cars make up 5 out of every 12 registered vehicles.

Step 4: Find the ratio of pickup trucks to total vehicles

Using the total number of parts \(12\), the ratio of pickup trucks to total vehicles is \(7:12\). This means that pickup trucks make up 7 out of every 12 registered vehicles.

Step 1: Determine the parts for each vehicle type

For the part - to - part ratio of passenger cars to pickup trucks (\(5:7\)), we will create a tape diagram. We will draw two tapes. One tape for passenger cars will be divided into 5 equal - sized parts, and the other tape for pickup trucks will be divided into 7 equal - sized parts. The length of each small part in both tapes should be the same to represent the proportional relationship.

Step 1: Analyze the part - to - part ratio

The part - to - part ratio of passenger cars to pickup trucks is \(5:7\). To find the number of equal - sized parts in the tape diagram, we add the number of parts representing passenger cars and the number of parts representing pickup trucks.

Step 2: Calculate the total number of parts

Using the formula \(5 + 7\), we get \(12\).

Answer:

1. \(5:7\) - Ratio of passenger cars to pickup trucks (for every 5 passenger cars, there are 7 pickup trucks).
2. \(7:5\) - Ratio of pickup trucks to passenger cars (for every 7 pickup trucks, there are 5 passenger cars).
3. \(5:12\) - Ratio of passenger cars to total vehicles (5 out of 12 total vehicles are passenger cars).
4. \(7:12\) - Ratio of pickup trucks to total vehicles (7 out of 12 total vehicles are pickup trucks).

Part b